A Note on the Representation Type of Pointed Irreducible Coalgebras and Unipotent Algebras

1977 ◽  
Vol 29 (1) ◽  
pp. 220-223
Author(s):  
David Trushin
Keyword(s):  
Finite Sequence ◽  
Representation Type ◽  
Finite Dimensional ◽  
Bounded Representation ◽  
Divided Powers

In this paper the representation type of the class of pointed irreducible coalgebras is studied. We refer the reader to [4] for the basic definitions. A coalgebra is of bounded representation type if there is a bound on the dimension of finite dimensional indecomposable comodules. In Section 1, we show that the representation type is dependent upon the size of the space of primitives. Indeed, a pointed irreducible coalgebra is of bounded type if and only if it is finite dimensional and the space of primitives is onedimensional, i.e. if and only if it is a coalgebra spanned by a finite sequence of divided powers.

scholarly journals Algebras of bounded finite dimensional representation type

1995 ◽  
Vol 37 (3) ◽  
pp. 289-302 ◽  
Author(s):  
Allen D. Bell ◽  
K. R. Goodearl
Keyword(s):  
Representation Type ◽  
Dimensional Representation ◽  
Dimensional Algebra ◽  
Finite Dimensional Representation ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Left And Right ◽  
Bounded Representation ◽  
Algebra Over A Field

It is well known that for finite dimensional algebras, “bounded representation type” implies “finite representation type”; this is the assertion of the First Brauer-Thrall Conjecture (hereafter referred to as Brauer-Thrall I), proved by Roiter [26] (see also [23]). More precisely, it states that if R is a finite dimensional algebra over a field k, such that there is a finite upper bound on the k-dimensions of the finite dimensional indecomposable right R-modules, then up to isomorphism R has only finitely many (finite dimensional) indecomposable right modules. The hypothesis and conclusion are of course left-right symmetric in this situation, because of the duality between finite dimensional left and right R-modules, given by Homk(−, k). Furthermore, it follows from finite representation type that all indecomposable R modules are finite dimensional [25].

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

scholarly journals Fields of Definition for Representations of Associative Algebras

2018 ◽  
Vol 62 (1) ◽  
pp. 291-304
Author(s):  
Dave Benson ◽  
Zinovy Reichstein
Keyword(s):  
Finite Extension ◽  
Representation Type ◽  
Essential Dimension ◽  
Extension Field ◽  
Associative Algebras ◽  
Separable Extension ◽  
Finite Representation ◽  
Field Of Definition ◽  
Finite Dimensional ◽  
A Finite Group

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

scholarly journals The class of(1,3)-groups with homocyclic regulator quotient of exponentp4has bounded representation type

2014 ◽  
Vol 400 ◽  
pp. 43-55 ◽  
Author(s):  
David M. Arnold ◽  
Adolf Mader ◽  
Otto Mutzbauer ◽  
Ebru Solak
Keyword(s):  
Representation Type ◽  
Bounded Representation

Hereditary artinian rings of finite representation type and extensions of simple artinian rings

1987 ◽  
Vol 102 (3) ◽  
pp. 411-420 ◽  
Author(s):  
Aidan Schofield
Keyword(s):  
Artinian Ring ◽  
Representation Type ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Finite Dimensional ◽  
Coxeter Diagram ◽  
Skew Fields ◽  
Left And Right ◽  
Natural Way

In [1], Dowbor, Ringel and Simson consider hereditary artinian rings of finite representation type; it is known that if A is an hereditary artinian algebra of finite representation type, finite-dimensional over a field, then it corresponds to a Dynkin diagram in a natural way; they show that an hereditary artinian ring of finite representation type corresponds to a Coxeter diagram. However, in order to construct an hereditary artinian ring of finite representation type corresponding to a Coxeter diagram that is not Dynkin, they show that it is necessary though not sufficient to find an extension of skew fields such that the left and right dimensions are both finite but are different. No examples of such skew fields were known at the time. In [3], I constructed such extensions, and the main aim of this paper is to extend the methods of that paper to construct an extension of skew fields having all the properties needed to construct an hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5).

THE CLASS OF (2, 2)-GROUPS WITH HOMOCYCLIC REGULATOR QUOTIENT OF EXPONENT  HAS BOUNDED REPRESENTATION TYPE

2014 ◽  
Vol 99 (1) ◽  
pp. 12-29
Author(s):  
DAVID M. ARNOLD ◽  
ADOLF MADER ◽  
OTTO MUTZBAUER ◽  
EBRU SOLAK
Keyword(s):  
Representation Type ◽  
Bounded Representation ◽  
Decomposable Groups

The class of almost completely decomposable groups with a critical typeset of type$(2,2)$and a homocyclic regulator quotient of exponent $p^{3}$is shown to be of bounded representation type. There are only$16$isomorphism at$p$types of indecomposables, all of rank $8$or lower.

Sign in / Sign up

Export Citation Format

Share Document